Symmetric and Skew Symmetric Matrices: Full Class 12 CBSE Guide

After mastering the transpose, the next major concept in NCERT Chapter 3 is symmetric and skew symmetric matrices. These are special types of square matrices defined entirely by their relationship to their own transpose. Understanding them unlocks two important theorems — and a standard 5-mark board question type where you express any square matrix as the sum of a symmetric and a skew symmetric matrix.

This guide covers both definitions, key properties of diagonal elements, two fundamental theorems, and a full worked example.

Definition of Symmetric Matrices

A square matrix A = [aij] is called symmetric if its transpose equals itself:

A' = A,  equivalently  aij = aji  for all i and j

In a symmetric matrix, every element reflected across the main diagonal has the same value. The matrix is “mirror symmetric” about its diagonal.

Example of a Symmetric Matrix

A = | √3    2    3  |
    |  2   -1.5 -1  |
    |  3   -1    1  |

Here, a₁₂ = a₂₁ = 2, a₁₃ = a₃₁ = 3, and a₂₃ = a₃₂ = −1. Taking A’ gives exactly A, confirming it is symmetric.

Quick Check for Symmetry

For a 2 × 2 matrix [[a, b], [c, d]], symmetry requires b = c. For a 3 × 3 matrix, the three pairs of off-diagonal elements must match across the diagonal.

A' = -A,  equivalently  aji = -aij  for all i and j

Every element reflected across the diagonal is the negative of its mirror element.

Critical Property: Diagonal Elements Are Always Zero

Setting i = j in the skew symmetric condition: aii = −aii, which means 2aii = 0, so aii = 0 for all i.

Every diagonal element of a skew symmetric matrix is zero. This is a key fact tested directly in CBSE exams.

Example of a Skew Symmetric Matrix

B = |  0    e    f |
    | -e    0    g |
    | -f   -g    0 |

Here B’ = −B, and all diagonal elements are 0.

Theorem 1: A + A’ is Symmetric and A − A’ is Skew Symmetric

For any square matrix A with real entries:

• A + A’ is always symmetric

• A − A’ is always skew symmetric

Proof for A + A’

Let B = A + A’. Then:

B' = (A + A')' = A' + (A')' = A' + A = A + A' = B

Since B’ = B, the matrix A + A’ is symmetric. ✓

Proof for A − A’

Let C = A − A’. Then:

C' = (A - A')' = A' - (A')' = A' - A = -(A - A') = -C

Since C’ = −C, the matrix A − A’ is skew symmetric. ✓

This theorem is proved step by step in NCERT and is a standard 4-mark proof question in CBSE boards.

Theorem 2: Every Square Matrix is a Sum of Symmetric and Skew Symmetric Matrices

Any square matrix A can be written as:

A = (1/2)(A + A') + (1/2)(A - A')

The first part, P = ½(A + A’), is symmetric (by Theorem 1 and the scalar property of transpose). The second part, Q = ½(A − A’), is skew symmetric.

This means every square matrix can be uniquely expressed as the sum of a symmetric and a skew symmetric matrix — a beautifully elegant result.

Worked Example: Expressing a Matrix as a Sum

Example (NCERT Example 22): Express B = [[2, −2, −4], [−1, 3, 4], [1, −2, −3]] as the sum of a symmetric and a skew symmetric matrix.

Step 1: Find B’

B' = |  2  -1   1 |
     | -2   3  -2 |
     | -4   4  -3 |

Step 2: Compute P = ½(B + B’)

B + B' = |  4  -3  -3 |     P = |  2    -3/2  -3/2 |
         | -3   6   2 |         | -3/2   3     1   |
         | -3   2  -6 |         | -3/2   1    -3   |

Verify P’ = P ✓ (symmetric)

Step 3: Compute Q = ½(B − B’)

B - B' = |  0  -1  -5 |     Q = |  0    -1/2  -5/2 |
         |  1   0   6 |         |  1/2   0     3   |
         |  5  -6   0 |         |  5/2  -3     0   |

Verify Q’ = −Q ✓ (skew symmetric, note all diagonal elements are 0)

Step 4: Confirm P + Q = B ✓

Additional Results Involving Symmetric and Skew Symmetric Matrices

Beyond the two main theorems, NCERT presents several additional results that are tested in CBSE board exams:

Result 1: If A and B are symmetric matrices of the same order, then AB − BA is skew symmetric.

Proof: (AB − BA)’ = (AB)’ − (BA)’ = B’A’ − A’B’ = BA − AB = −(AB − BA). Since the transpose equals the negative, AB − BA is skew symmetric. ✓

Result 2: The matrix B’AB is symmetric if A is symmetric, and skew symmetric if A is skew symmetric.

Proof: (B’AB)’ = B’A’(B’)’ = B’A’B. If A’ = A (symmetric), this equals B’AB — symmetric. If A’ = −A (skew symmetric), this equals −B’AB — skew symmetric.

These results carry 4 marks each in CBSE boards and the proof structure is always the same: compute the transpose, apply the given condition on A, and show the result equals ±the original.

Key Results to Memorise

• AB − BA is skew symmetric whenever A and B are symmetric matrices of the same order.

• If A is symmetric: (A + A’) = 2A (already symmetric) and A − A’ = O.

• If A + A’ = I, the value of α in [[cosα, −sinα], [sinα, cosα]] satisfies a specific condition — a standard MCQ in CBSE.

CBSE Board Exam Tips

• Diagonal zero: In any skew symmetric matrix, ALL diagonal elements must be 0. If a question gives a skew symmetric matrix with a diagonal entry that isn’t 0, something is wrong.

• Symmetric check: For A to be symmetric, verify A’ = A by inspection — compare elements across the diagonal.

• 5-mark question format: Express a 3 × 3 matrix as sum → compute P and Q separately → verify each → state P + Q = original matrix.

• Theorem proofs: Be able to prove both Theorem 1 cases using transpose properties.

Suggested Reading on ChampionsPrep

• Transpose of a Matrix: Properties, Reversal Law and Class 12 Examples

• Types of Matrices: All Seven Types — Definitions and Examples

• Invertible Matrices: Definition and Theorems for CBSE Class 12

Test Your Knowledge

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Symmetric and skew symmetric matrices generate some of the highest-value questions in CBSE boards — proofs, verifications, and the sum-decomposition method. Master these with targeted practice.

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